Simplify and expand the following expression: $ \dfrac{5}{2x + 9}+\dfrac{x - 8}{x - 10} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2x + 9)(x - 10)$ Multiply the first term by $\dfrac{x - 10}{x - 10}$ $ \begin{align*} \dfrac{5}{2x + 9} \times \dfrac{x - 10}{x - 10} & = \dfrac{(5)(x - 10)}{(2x + 9)(x - 10)} \\ & = \dfrac{5x - 50}{(2x + 9)(x - 10)}\end{align*} $ Multiply the second term by $\dfrac{2x + 9}{2x + 9}$ $ \begin{align*} \dfrac{x - 8}{x - 10} \times \dfrac{2x + 9}{2x + 9} & = \dfrac{(x - 8)(2x + 9)}{(x - 10)(2x + 9)} \\ & = \dfrac{2x^2 - 7x - 72}{(x - 10)(2x + 9)}\end{align*} $ Now we have: $ = \dfrac{5x - 50}{(2x + 9)(x - 10)} + \dfrac{2x^2 - 7x - 72}{(x - 10)(2x + 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5x - 50 + 2x^2 - 7x - 72}{(2x + 9)(x - 10)} $ $ = \dfrac{-2x - 122 + 2x^2}{(2x + 9)(x - 10)}$ Expand the denominator: $ = \dfrac{-2x - 122 + 2x^2}{2x^2 - 11x - 90}$